可分B-空间中一般加权和的大数定律

可分B-空间中一般加权和的大数定律 A Law of Large Numbers for General Weighted Sums in a Separable B-Space first-author 刘吉定, 男, 副教授, 主要从事概率论极限理论方面的研究. E-mail:jidingliu@163.com 刘吉定, 男, 副教授, 主要从事概率论极限理论方面的研究. E-mail: jidingliu@163.com jidingliu@163.com 武汉工程大学 理学院, 湖北 武汉 430074 武汉工程大学 理学院, 湖北 武汉 430074 School of Sciences, Wuhan Institute of Technology, Wuhan 430074, Hubei, China School of Sciences, Wuhan Institute of Technology, Wuhan 430074, Hubei, China 利用可分Banach空间中已有的概率不等式及对称化方法, 研究了可分B-空间中加权系数具有某些较弱性质的加权和收敛问题, 得到了B-值独立同分布随机元序列的这类加权和的强、弱大数定律成立的充分条件, 对可分Banach空间中的Cesaro大数定律和欧拉大数定律进行了推广.同时, 得到了实值独立同分布随机变量序列的这类加权和的强、弱大数定律成立的充分条件. By the probability inequality and the symmetrization method obtained in a separable Banach space, convergence for the weighted sums whose weighting coefficients have some weaker natures in a separable Banach space is investigated, and the sufficient condition is obtained for the strong and weak law of large numbers of the weighted sums of a sequence of B-valued random elements that are independent and identically distributed, and the Cesaro law of large numbers and the Euler law of large numbers in a separable Banach space are spread.At the same time, the sufficient condition is obtained for the strong and weak law of large numbers of the weighted sums of a sequence of real-valued random variables that are independent and identically distributed. 无穷维空间 随机元的加权和 几乎处处收敛 an infinite-dimensional space weighted sums of random elements almost sure convergence 引言 1 概率论中极限定理的研究一直受到学者们的重视 [ 1 1 ~ 3 3 1~3 ] ，Heinkel [ 4 4 ] 研究了可分B-空间中的Cesaro大数定律，Bozorgnia和Rao [ 5 5 ] 研究了可分B-空间中的欧拉大数定律.加权系数为一般情形的加权大数定律与加权系数为特殊情形的加权大数定律相比较，前者由于其所具有的广泛性，在理论上显得更加重要.本文按照文献[ 4 4 ]的思想方法对可分Banach空间中一般加权和的大数定律进行了研究，得到了可分B-空间中这类加权和的强、弱大数定律成立的充分条件.对Heinkel [ 4 4 ] , Bozorgnia, Rao [ 5 5 ] 以及刘吉定 [ 6 6 ] 等人所作的工作进行了推广，给出了这些结果的统一形式.作为推论，实值i.i.d随机变量序列相应的一般加权大数定律自然成立. 设( Ω , F , P )是一概率空间， B 是可分的实Banach空间，{ X k } k ≥0 是定义在( Ω , F , P )上的独立同分布B-值随机元序列，{ X k ′} k ≥0 是{ X k } k ≥0 的独立版本， K 表示常数，在某些地方，它可能与 α 有关，在不同的地方，它有可能表示不同的常数. 主要结果及其证明 1 定理1   设{ λ nk }为一非负实双重数组， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459417&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459417&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459417&type=middle =1，且存在单调增加的正数列{ μ n }，使得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459330&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459330&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459330&type=middle 若 E ‖ X 0 ‖＜∞，则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459335&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459335&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459335&type=middle 证   不妨假设 EX 0 =0(否则用 X k － EX k 取代 X k ).因为 B 是可分的Banach空间，故存在 B 的可数稠子集{ x j : j =1, 2, …}.∀ ε ＞0(不妨认为 ε ≤1/4)，∀ j ∈{1, 2, …}，定义 E j ={ x :‖ x － x j ‖≤ ε }，记 F 1 = E 1 , …, http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459421&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459421&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459421&type=middle , …; Y k = http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459426&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459426&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459426&type=middle . 由于{ x j : j =1, 2, …}是 B 的可数稠子集，所以∀ ω ∈ Ω , ∀ k ≥0，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459340&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459340&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459340&type=middle 根据 Y 0 及{ F j } j ≥1 的定义，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459345&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459345&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459345&type=middle 由于{ X k } k ≥0 是同分布序列，所以根据(3)式，存在自然数 N = N ( ε )，使得∀ k ≥0，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459350&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459350&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459350&type=middle 从而 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459354&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459354&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459354&type=middle 根据 Y k 的定义，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459359&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459359&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459359&type=middle 由(2)式，知∀ n ≥0, ∀ ω ∈ Ω ，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459363&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459363&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459363&type=middle 对于每一个 j ∈{1, 2, …, N }，显然{ I { X k ∈ F j } } k ≥0 为实值i.i.d随机变量序列，且 E | I { X 0 ∈ F j } |＜∞，所以根据文献[ 7 7 ]定理1，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459368&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459368&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459368&type=middle 由此可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459374&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459374&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459374&type=middle 由 Y 0 的定义， EX 0 =0，(2)式及{ X k } k ≥0 是同分布序列及(4)式，知 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459378&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459378&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459378&type=middle 由(5)~(7)、(9)式及Chebyshev不等式可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459385&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459385&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459385&type=middle 由(10)式及(8)式，知 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459390&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459390&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459390&type=middle 定理1证毕. 对每个常数 β ＞－1，定义 [ 4 4 ] ： A 0 β =1, A 1 β = β +1, …, A k β =( β +1)…( β + k )/ k !, …. 推论1   若 E ‖ X 0 ‖＜∞，则对任意的 α ∈(0, 1)，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459428&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459428&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459428&type=middle . 证   由定理1及文献[ 4 4 ]中引理1即得到. 推论1就是文献[ 4 4 ]中引理3. 推论2   若 E ‖ X 0 ‖＜∞，则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459394&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459394&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459394&type=middle 证   由定理1及文献[ 8 8 ]中引理1即得. 推论2就是文献[ 6 6 ]中定理. 定理1对文献[ 4 4 , 6 6 ]中的相应结果进行了统一推广. 推论3   设{ X k } k ≥0 是定义在( Ω , F , P )上的实值i.i.d随机变量序列，{ λ nk }为一非负实双重数组， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459433&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459433&type=middle ，且存在单调增加的正数列{ μ n }，使 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459438&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459438&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459438&type=middle .若 E | X 0 |＜∞，则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459442&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459442&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459442&type=middle . 证   由定理1立即得到推论3. 定理2   设{ λ nk }为一非负实双重数组， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459448&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459448&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459448&type=middle .若 E ‖ X 0 ‖ 1/ α ＜∞，则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459400&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459400&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459400&type=middle 证   利用Hoffmann-Jogensen不等式 [ 9 9 ] 及指数不等式 [ 10 10 ] 可以完成定理2的证明. 推论4    http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459451&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459451&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459451&type=middle .成立的充要条件是 E ‖ X 0 ‖ 2 ＜∞. 证   由定理2, 文献[ 8 8 ]中引理1及文献[ 5 5 ]中定理2.3(ⅱ)的必要性的证明过程即可得到推论4. 推论4就是文献[ 5 5 ]中定理2.3(ⅱ). 推论5    http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459454&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459454&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459454&type=middle .成立的充要条件是 E ‖ X 0 ‖ 1/ α ＜∞. 证   由定理2、文献[ 4 4 ]中引理1及文献[ 4 4 ]中定理2必要性的证明过程即可得到推论5. 推论5就是文献[ 4 4 ]中定理2. 由此可见，定理2对文献[ 4 4 , 5 5 ]中的相应结果进行了统一推广. 推论6   设{ X k } k ≥0 是定义在( Ω , F , P )上的实值i.i.d随机变量序列，{ λ nk }为一非负实双重数组， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459458&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459458&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459458&type=middle .若 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459404&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459404&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459404&type=middle 则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459410&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459410&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459410&type=middle 证   由定理2立即得到推论6. 下面给出经验过程中的一个结果.有关记号见文献[ 11 11 ]. 推论7   设 S =[0, 1], G ⊂ C [0, 1], ∀ f , g ∈ G , http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459464&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459464&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459464&type=middle 完全有界， X 为( Ω , F , P )上取值于可测空间( S , A )中的随机元.{ X k } k ≥0 是 X 的独立版本序列，则对于任何 α ∈(0, 1)， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459467&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459467&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459467&type=middle .成立的充要条件是 E ‖ f ( X )‖ G 1/ α ＜∞. 证   由( G , d )完全有界，可知： G 是可数决定族.且∀ x ∈ S , ‖ f ( x )‖ G ＜∞.记随机映射 X ( f ): G → R 的值空间为 Q .令 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459470&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459470&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459470&type=middle .显然 Q ⊂ R .由推论5知：为证推论7，只需证明 R 可分. 令{ x i } i ≥1 是[0, 1]中的一切有理数.由( G , d )完全有界，容易看出：∀ x ∈ S , ∀ ε ＞0, ∃ x i 使得‖ f ( x )－ f ( x i )‖ G ＜ ε .故 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459474&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459474&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459474&type=middle : G → R | i =1, 2, …}为 R 的可数稠子集，从而 R 可分.推论7得证. 参考文献 NADERI H, AMINI M, BOZORGNIA A.Complete convergence for weighted sums of weakly negative dependent of random variables[J]. 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